
Note that the lattice spacing of the Dirac-delta peaks in the momentum direction is twice that of the position direction in the Wigner picture. For a clear diagram of the phase space unit cell distribution, see Fig. 1a of [1]}. Treating each \(\delta \) -function as a Gaussian of infinitely small width in phase space, we see that the effect of the noise channel is to replace the \(\delta \) -functions with Gaussian distributions with covariance \(\mathbf {Y}_{\rm gkp}\) , by using eq:noiseapp. Thus, the linear combination of \(\delta \) -functions is mapped to a linear combination of Gaussian functions centred at the same points in phase space and with the same weights in the linear combination of eq:idealwigner.
